Question

True or False The graphs of \(y=\sin x\) and \(y=\cos x\) are identical except for a horizontal shift.

Short answer

True. The graphs of \(y = \sin x\) and \(y = \cos x\) are identical except for a horizontal shift of \(\frac{\pi}{2}\) radians.

Step-by-step solution

Understanding the Basic Graphs

First, denote the basic shapes of the sine and cosine graphs. The graph of \(y = \sin x\) is a wave that starts at the origin (0,0), oscillates between -1 and 1, and repeats every \(2\pi\) radians. The graph of \(y = \cos x\) also oscillates between -1 and 1 and repeats every \(2\pi\). However, it starts at (0,1).

Comparing Sine and Cosine Graphs

Identify the differences and similarities between the two graphs. Notice that \(y = \cos x\) can be derived by shifting the graph of \(y = \sin x\) horizontally to the left by \(\frac{\pi}{2}\) radians. Mathematically, this relationship can be expressed as \[ \cos x = \sin \left( x + \frac{\pi}{2} \right) \].

Concluding the Relationship

Based on the observations in Steps 1 and 2, it can be concluded that the graphs of \(y = \sin x\) and \(y = \cos x\) are indeed identical except for the horizontal shift of \(\frac{\pi}{2}\) radians. This confirms that the statement is true.

Key concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. Its graph, denoted as \( y = \sin x \), resembles a smooth wave that oscillates between -1 and 1. This oscillation repeats every \( 2\pi \) radians, so each complete wave cycle covers an interval of \( 2\pi \). The graph starts at the origin (0,0), rises to a peak of 1, returns to 0, dips to -1, and then comes back to 0, completing one full cycle.

Key characteristics:

• Oscillates between -1 and 1
• Repeats every \( 2\pi \) radians
• Starts at the origin (0,0)

These properties provide a foundation for understanding how the sine function behaves and how it can be transformed.

Cosine Function

The cosine function, denoted as \( y = \cos x \), shares many similarities with the sine function. Its graph also oscillates between -1 and 1 and repeats every \( 2\pi \) radians. However, instead of starting at the origin, the cosine function starts at (0,1). This means its initial value is 1 when \( x = 0 \).

Key characteristics:

• Oscillates between -1 and 1
• Repeats every \( 2\pi \) radians
• Starts at (0,1)

While the shapes and behaviors of the cosine and sine functions are similar, their starting points differ. This difference is essential for understanding the horizontal shift between the graphs of these two functions.

Horizontal Shift

A horizontal shift in a graph involves moving the graph left or right along the x-axis without altering its shape. For the graphs of the sine and cosine functions, this shift is crucial in understanding their relationship.

To visualize this:

• The graph of \( y = \cos x \) can be obtained by shifting the graph of \( y = \sin x \) to the left by \( \frac{\pi}{2} \) radians.
• Mathematically, this shift is represented as: \[ \cos x = \sin \( x + \frac{\pi}{2} \) \]

This suggests that if you start with the sine function graph and move it left by \( \frac{\pi}{2} \) radians, you'll get the cosine function graph. This horizontal shift showcases one of the most intriguing relationships in trigonometry. Understanding this principle allows for better comprehension of how different trigonometric functions interrelate.

Horizontal shifts are not only limited to trigonometric functions. They apply to many other types of functions and are a fundamental concept in graph transformations.